2\pi \int _{-3}^3\left(e^{2x}+e^{-2x}\right)\sqrt{1+\frac{e^{-4x}+\left(e^{4x}-1\right)^2}{4}}\:

To solve the given integral expression:

\[
2\pi \int_{-3}^3 \left(e^{2x} + e^{-2x}\right) \sqrt{1 + \frac{e^{-4x} + \left(e^{4x} - 1\right)^2}{4}} \, dx
\]

### Step 1: Simplify the expression inside the square root

First, let's simplify the expression inside the square root:

\[
1 + \frac{e^{-4x} + \left(e^{4x} - 1\right)^2}{4}
\]

Expand \(\left(e^{4x} - 1\right)^2\):

\[
\left(e^{4x} - 1\right)^2 = e^{8x} - 2e^{4x} + 1
\]

So the expression becomes:

\[
1 + \frac{e^{-4x} + e^{8x} - 2e^{4x} + 1}{4}
\]

Simplify further:

\[
\frac{e^{-4x} + e^{8x} - 2e^{4x} + 1}{4} = \frac{1}{4}\left(e^{-4x} + e^{8x} - 2e^{4x} + 1\right)
\]

Add 1 to this:

\[
1 + \frac{1}{4}\left(e^{-4x} + e^{8x} - 2e^{4x} + 1\right) = \frac{4}{4} + \frac{e^{-4x} + e^{8x} - 2e^{4x} + 1}{4} = \frac{4 + e^{-4x} + e^{8x} - 2e^{4x} + 1}{4} = \frac{5 + e^{-4x} + e^{8x} - 2e^{4x}}{4}
\]

Therefore, the expression inside the square root becomes:

\[
\sqrt{\frac{5 + e^{-4x} + e^{8x} - 2e^{4x}}{4}}
\]

### Step 2: Substitute and simplify

Substituting back into the integral:

\[
2\pi \int_{-3}^3 \left(e^{2x} + e^{-2x}\right) \cdot \frac{\sqrt{5 + e^{-4x} + e^{8x} - 2e^{4x}}}{2} \, dx
\]

This simplifies to:

\[
\pi \int_{-3}^3 \left(e^{2x} + e^{-2x}\right) \sqrt{5 + e^{-4x} + e^{8x} - 2e^{4x}} \, dx
\]

### Step 3: Consider symmetry and possible further simplifications

Notice that the integrand is symmetric with respect to \(x = 0\). This can help in evaluating the integral or finding further simplifications. Given the complexity, if the integral is not solvable analytically, a numerical approach may be necessary.

### Step 4: Evaluate the integral

Given the complexity of the integrand, it's unlikely that it simplifies to an elementary function. At this point, numerical integration may be the best approach.

Would you like to proceed with a numerical approximation, or do you have specific conditions or simplifications in mind?

Learn how to evaluate the integral 2π ∫(-3)^3 (e^(2x) + e^(-2x)) * sqrt(1 + (e^(-4x) + (e^(4x) - 1)^2)/4) dx. This involves simplifying the integrand, understanding properties of exponential and square root functions, and possibly using numerical methods for evaluation. Suitable for undergraduate students studying integral calculus and advanced functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation